One underdeterminate epistemic possibility in this connection is that a theory about perception itself might involve making a judgment about the cardinality of the set of real numbers. The open coloring question, for example, might conceivably be relevant to a theory of color perception (albeit quite in abstracto), and since two samples of open-coloring axioms in tandem resolve the natural powerset to ℵ2, one might reason from an attendant theory of color perception (if it's actually possible/relevant) to the conclusion that the natural powerset is ℵ2.
Another (not-so-clear) possibility might be that forcing as a mathematical phenomenon has a physical counterpart, and this counterpart can change the size of physical continua. I've been working on an attempt to model laws-of-physics on infinite conjunctions in infinitary logic where for t (time) = n, our physical universe has an infinitary logical signature ℒ(ωf(n), ωg(n)) such that some functions f and g yield an evolution of the physical world's logical signature over time (this is to try to implement Lee Smolin's changing-laws-of-physics idea), but to be honest, I haven't gotten much further than a nifty background for a science fiction storyline, not a genuine scientific hypothesis. (I.e., what predictive value does this "model" have, if any? The best I've thought of would be that our capacity for continuous perception would change with the changing cardinality of physical continua, but how would that be "noticeable" or evaluable, then?)
Broadly, one problem with thinking that mathematical physics, much less experimental physics that's mathematically informed, would be amenable to novel reasoning for or against CH is that there are an extremely vast number of alternatives to CH in higher set theory. ZFCwise, there are absolutely infinitely many alephs that the natural powerset can be forced to equal, and beyond the edge of ZFC, there is even the possibility of forcing the Continuum to equal absolute infinity itself (see also Timothy James' essay on predicativism in the philosophy of mathematics on the "indefinite extensibility" of the natural powerset; or consider that the surreal number line itself contains absolutely infinitely many infinitesimals in every interval, incl. [0, 1]). Perhaps physics would at least allow us to eliminate the prior disjunct in {CH ∨ ~CH} but the latter disjunct is so internally vast that said elimination would be as meager a contribution to the issue as possible. Even worse, it's not only that the natural powerset can be forced to equal so many things on its own, but: the powerset of the zeroth aleph can be forced to equal the powerset of the first aleph, as well as the second, third, fourth, etc. alephs, and so indeed, modulo the proper-class scales of options, we might force every well-ordered transfinite cardinal, prior to ℶ1, to equal ℶ1, so that the first beth is a fixed point of the aleph function.ℝ On the surface, it is hard to say how empirical information, or mathematical models of said information, would include strong, clear reasons for filtering in, or out, so many options.
ℝEven more insidiously, suppose that the well-ordering principle is waived (because the basic axiom of choice, or whichever choice axiom, is waived). Then it is possible for the cardinality of the Continuum to be a transfinite cardinal, but not from the well-ordered sequence of such cardinals, i.e. it would not be an aleph but perhaps similar to (apparently not identical to, though) an amorphous set.
Perhaps most insidiously of all, suppose you waive the powerset axiom itself. You can still use the classical diagonal argument to show that the set of real numbers is not bijective with the set of natural numbers, but you no longer have that 2ℵ0 is the arithmetical expression of this difference. (I don't think this is plausible at all, since I think John Conway's explanation for the exponential expression fitting the case is a perfectly apparent explanation, but on the other hand, see again James' predicativist apologetics for how the exponential function on whichever X can come apart from the concept of "the set of all subsets of X.")
EDIT: Though this answer was accepted and has received a number of upvotes, it was also downvoted, and I feel like I worded it in an imprecise way. Firstly, on the "yes" side, my references to open coloring axioms and physical forcing are very speculative; insofar as reductionism is not in vogue anymore, I imagine that biological/neurological theories of color perception might indeed be relevant to physical theories in a way that could also play into evaluating some version of the Continuum problem, but I am not especially well-versed in actual physics, biology, or neurology, so I feel like I should emphasize just how speculative my comments on this score are.
Second, on the "maybe not" side: overall, I do not think that there is really just one powerset function. Cantor's theorem has it that various sets of subsets must exceed their bases, but complications involving definable vs. hyperdefinable (or even antidefinable) subsets seem to multiply the question of such a function. I think that the enduring desire to "settle" the Continuum issue is often derived from the seeming continuity involved in physical perception, so that, "How many points are there in physical space?" seems like a question of external/objective reality. So rather than say that every powerset function might be resolved by some future theory of physics, I would rather say that something like "the set of all physically realizable subsets of a countable set" would be the specific subtype of the powerset function whose size could be "settled" in such a theory. But this too is speculative, after all; I offer these conjectures as an answer to the OP question in only the bare sense that they address the modal term in that question per the title of the OP post: i.e., in some abstract sense of "can," CH, or a version of it anyway, "can" be settled by physics.
Finally, there is one more "insidious" variation on the "maybe not" theme that comes to my mind, one based on paraconsistent set theory. In PST, perhaps, one might force the Continuum's cardinality to equal several inconsistent numbers at once; applying this "model" to physics, or physics to this "model," would then mean bringing in considerations like a paraconsistent theory of superposition. Paraconsistent logic is motivated by the desire to avoid an inferential explosion, and forcing the Continuum's cardinality to equal every option otherwise delineated above is almost the same as (or maybe even identical to, eventually) such an explosion; so a paraconsistent set theorist would still be motivated to include some restrictions on their "insidious" standpoint, to rule out the most deviant alternative to CH of all.
But so again, overall, for a theory of physics to settle any version of the Continuum question would mean that such a theory would have to sort through questions about choice axioms, forcing, the geometry of perception, etc., all in a way that comports with how theories of physics are reliably set up. I'm not a physicist and I am reluctant to press my claims, here, too strongly, out of concern for veering off into pseudoscientific territory. I appreciate that my answer was accepted, and I think my answer involves informed reflection on the parameters of the OP question, but insofar as all this is philosophical reflection, I am still highly uncertain about my conclusions in this case.
